I am not sure how well it would work there but, presumably you are aware of iron-on edge banding and, don’t have any on-hand (or it’s too much hassle).
I used edge banding for this project:
I have been known to color edges of smaller plywood things with a Sharpie or paint pen (e.g., Uni Posca). The metallic sharpies can be particularly nice for things like ornaments. Much easier to avoid getting it on the other surfaces than applying something by brush.
+1 for @geek2nurse ’s suggestion. You can wrap a piece of shop/paper towel around the end of a finger and apply with that.